The components of the graphs D(n,q) provide the best-known general lower bound for the number of edges in a graph with n vertices and no cycles of length less than g. In this paper, we give a new, ...short, and simpler proof of the fact that the length of the shortest cycle appearing in D(n,q) is n+5 when n is odd, and n+4 when n is even.
Short rainbow cycles in graphs and matroids DeVos, Matt; Drescher, Matthew; Funk, Daryl ...
Journal of graph theory,
February 2021, 2021-02-00, 20210201, Volume:
96, Issue:
2
Journal Article
Peer reviewed
Open access
Let
G be a simple
n‐vertex graph and
c be a coloring of
E
(
G
) with
n colors, where each color class has size at least 2. We prove that
(
G
,
c
) contains a rainbow cycle of length at most
⌈
n
2
⌉, ...which is best possible. Our result settles a special case of a strengthening of the Caccetta‐Häggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also holds for cographic matroids, but fails for binary matroids.
Reed's Conjecture states that χ(G)≤⌈(Δ(G)+ω(G)+1)/2⌉, where χ(G), Δ(G) and ω(G) are the chromatic number, maximum degree and clique number of a graph G, respectively.
In this note, we prove this ...conjecture for maximal triangle-free graphs with maximum degree less than 7. Moreover, we show that Reed's Conjecture holds for all graphs with girth at least 5 up to at least 30 vertices and for all triangle-free graphs G up to at least 32 vertices such that χ(G)≠5 which improves similar results given in Jan Goedgebeur, On minimal triangle-free 6-chromatic graphs, J. Graph Theory, 93(2020), 34–48.
Objective
To systematically review and create nomograms of flaccid and erect penile size measurements.
Methods
Study key eligibility criteria: measurement of penis size by a health professional using ...a standard procedure; a minimum of 50 participants per sample. Exclusion criteria: samples with a congenital or acquired penile abnormality, previous surgery, complaint of small penis size or erectile dysfunction. Synthesis methods: calculation of a weighted mean and pooled standard deviation (sd) and simulation of 20 000 observations from the normal distribution to generate nomograms of penis size.
Results
Nomograms for flaccid pendulous n = 10 704, mean (sd) 9.16 (1.57) cm and stretched length n = 14 160, mean (sd) 13.24 (1.89) cm, erect length n = 692, mean (sd) 13.12 (1.66) cm, flaccid circumference n = 9407, mean (sd) 9.31 (0.90) cm, and erect circumference n = 381, mean (sd) 11.66 (1.10) cm were constructed. Consistent and strongest significant correlation was between flaccid stretched or erect length and height, which ranged from r = 0.2 to 0.6. Limitations: relatively few erect measurements were conducted in a clinical setting and the greatest variability between studies was seen with flaccid stretched length.
Conclusions
Penis size nomograms may be useful in clinical and therapeutic settings to counsel men and for academic research.
Background: The native goat populations in southern Odisha viz. Narayanpatna, Malkangiri and Ganjam were characterized phenotypically in the present study. Ganjam goats stood apart from other ...populations due to their large, pointed horns and stature. Odisha state has 30 districts and 10 different agro-climatic zones. Methods: The present study area is under North Eastern ghat and Eastern ghat high land agro climatic zones. An extensive management system is used to raise all the goats with a flock size varying from 6 to 45. The body weight was recorded from birth to 12 months at an interval of 3 months. Result: The average adult body weight from these 3 goat populations was 17.39±0.20, 16.31±0.17, 16.39±0.20 kg for Ganjam, Malkangiri and Narayanpatna goats, respectively revealing that Ganjam goats were heavier than the other two populations. The average age at sexual maturity and kidding interval was recorded as 318.24±5.24 and 312.06±1.28 days, 236.76±6.23 and 223.21±0.88 days, 218.34±4.28 and 218.22±0.76 days for Ganjam, Narayanpatna and Malkangiri goats, respectively revealing that Ganjam goats had significantly higher estimates than other two types. Males of all goat types were found to have higher body weight than the female counterparts beyond 3 months of age and found to be conspicuous at later age. This information on the physical characterization and production potentials may be helpful in developing further breed improvement strategies and breed registration.
A subgroup of the automorphism group of a graph Γ is said to be half-arc-transitive on Γ if its action on Γ is transitive on the vertex set of Γ and on the edge set of Γ but not on the arc set of Γ. ...Tetravalent graphs of girths 3 and 4 admitting a half-arc-transitive group of automorphisms have previously been characterized. In this paper we study the examples of girth 5. We show that, with two exceptions, all such graphs only have directed 5-cycles with respect to the corresponding induced orientation of the edges. Moreover, we analyze the examples with directed 5-cycles, study some of their graph theoretic properties and prove that the 5-cycles of such graphs are always consistent cycles for the given half-arc-transitive group. We also provide infinite families of examples, classify the tetravalent graphs of girth 5 admitting a half-arc-transitive group of automorphisms relative to which they are tightly-attached and classify the tetravalent half-arc-transitive weak metacirculants of girth 5.
For integer k≥2 and prime power q, the algebraic bipartite graph D(k,q) proposed by Lazebnik and Ustimenko (1995) is meaningful not only in extremal graph theory but also in coding theory and ...cryptography. This graph is q-regular, edge-transitive and of girth at least k+4. For its exact girth g=g(D(k,q)), Füredi et al. (1995) conjectured g=k+5 for odd k and q≥4. This conjecture was shown to be valid in 2016 when (k+5)/2 is the product of an arbitrary factor of q−1 and an arbitrary power of the characteristic of Fq. In this paper, we determine all the girth cycles of D(k,q) for 3≤k≤5, q>3, and those for 3≤k≤14, q=3.
Let Φ=(Γ,φ) be a T-gain graph or a complex unit gain graph and A(Φ) be its adjacency matrix. The rank of (Γ,φ) is denoted by r(Γ,φ) which is the rank of its adjacency matrix. If the underlying graph ...Γ of Φ has at least one cycle, then the girth of Φ is denoted by gr(Γ) or simply by gr, which is the length of the shortest cycle in Γ. In this paper, we prove r(Γ,φ)≥gr−2 for a T-gain graph (Γ,φ). Moreover, we characterize T-gain graphs satisfy r(Γ,φ)=gr−2 and r(Γ,φ)=gr.
PDF
